Integration by Substitution

Integration by substitution relies on clear pattern recognition and precise algebraic manipulation, both of which are strengthened by active, collaborative practice. Students need immediate feedback to correct substitution choices and limit conversions, which live practice through card sorts, relays, and error hunts provides efficiently.

• Year 13
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Identify the inner function and its derivative within a composite function to determine an appropriate substitution.
2. Calculate the differential of the substitution (du) and adjust the differential of the integration variable (dx).
3. Transform definite integrals by changing the limits of integration to correspond to the new variable of substitution.
4. Construct the integrated form of a composite function by substituting u and du, then integrating with respect to u.
5. Evaluate definite integrals using substitution, either by converting limits or by back-substituting to the original variable.

Decision Matrix

⏱ 30 min·Pairs

Card Sort: Substitution Matches

Create cards showing original integrals, possible u choices, du expressions, and simplified forms. Students in pairs sort and match complete sets, then test by integrating a sample. Class shares one challenging match.

Prepare & details

Explain the process of choosing an appropriate substitution for integration.

Facilitation Tip: For Card Sort: Substitution Matches, set a timer and have pairs justify each match aloud before moving to the next set, reinforcing the connection between g(x) and g'(x) dx.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Decision Matrix

⏱ 40 min·Small Groups

Relay Challenge: Definite Integrals

Divide class into small groups lined up at board. First student chooses u, writes du and new limits; next integrates; third verifies value. Groups compete for fastest correct solution, then discuss strategies.

Prepare & details

Justify why a change of variables requires a corresponding change in limits for definite integrals.

Facilitation Tip: During Relay Challenge: Definite Integrals, circulate with a checklist to verify limit conversions at each station and intervene immediately if teams skip this step.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Decision Matrix

⏱ 35 min·Small Groups

Error Hunt: Worked Examples

Provide printed sheets with five substitution solutions containing common errors like missing du or wrong limits. Small groups identify issues, correct them, and explain fixes to the class.

Prepare & details

Construct an integral solution using the method of substitution.

Facilitation Tip: In Error Hunt: Worked Examples, require students to rewrite each corrected step fully rather than just circling errors, to rebuild procedural memory.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Decision Matrix

⏱ 25 min·Pairs

Problem Creation: Peer Swap

Individuals craft two original integrals needing substitution, swap with partners, solve each other's, then check solutions together using graphing software if available.

Prepare & details

Explain the process of choosing an appropriate substitution for integration.

Facilitation Tip: In Problem Creation: Peer Swap, insist that the creator writes a clear solution key before swapping, so peers can compare their work directly to a standard.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Teaching This Topic

Teachers should model substitution as a reversible process—first in differentiation, then in integration—so students see the parallel. Avoid rushing to the final answer; emphasize the intermediate step of expressing dx in terms of du to prevent missing factors. Research shows students benefit from seeing multiple correct substitutions for the same integral, so compare approaches during whole-class discussion to normalize flexibility.

What to Expect

Students will confidently choose u and compute du, convert limits accurately, and integrate transformed expressions without unnecessary back-substitution. They will explain their choices and justify steps to peers, demonstrating both procedural fluency and conceptual understanding of the chain rule reversal.

Watch Out for These Misconceptions

Common Misconception

During Relay Challenge: Definite Integrals, watch for students who keep the original x-limits instead of converting them to u-values.

What to Teach Instead

At each station, ask teams to write both the u-expression and the converted limits before computing the integral, then check that their numerical result matches the original definite integral’s approximate value.

Common Misconception

During Card Sort: Substitution Matches, watch for students who pair f(g(x)) with g(x) without identifying g'(x) dx as a separate factor.

What to Teach Instead

Require pairs to sort by u = g(x) first, then match g'(x) dx separately to f(g(x)) to form complete integrands; unmatched cards signal missing du.

Common Misconception

During Error Hunt: Worked Examples, watch for students who back-substitute unnecessarily for definite integrals, adding extra steps.

What to Teach Instead

Ask students to circle any step that returns to x and discuss aloud why this is unnecessary; the hunt sheet should show the integral solved directly in u terms.

Assessment Ideas

Quick Check

After Card Sort: Substitution Matches, circulate to listen for pairs explaining their u choice and du in terms of x. Ask one pair to present their match and justification to the class, noting whether they correctly identified both u and the required dx substitution.

Exit Ticket

After Relay Challenge: Definite Integrals, collect each team’s final answer and their u-limits table. Assess both the numerical result and the accuracy of limit conversion; return feedback before the next lesson.

Peer Assessment

During Problem Creation: Peer Swap, have peers use a rubric to score the creator’s clarity of steps, correct substitution, and limit conversion. Swap roles so both students give and receive feedback on the same problem.

Extensions & Scaffolding

• Challenge: Provide integrals that require two substitutions, such as ∫ x^3 e^(x^4) dx, and ask students to write a step-by-step guide for peers.
• Scaffolding: Give pre-filled substitution tables with partial du values so students focus only on choosing u and rewriting limits.
• Deeper exploration: Explore integrals where substitution and parts both apply, such as ∫ x e^(2x) dx, and compare efficiency of each method.

Key Vocabulary

Substitution	Replacing a part of an expression with a new variable, typically 'u', to simplify the integration process.
Differential	The infinitesimal change in a variable, represented as 'du' or 'dx', which is crucial for substitution in integration.
Limits of Integration	The upper and lower bounds of a definite integral, which must be adjusted when a substitution changes the variable of integration.
Composite Function	A function that is created by applying one function to the result of another function, often in the form f(g(x)).